reserve V for RealLinearSpace,
  u,u1,u2,v,v1,v2,w,w1,x,y for VECTOR of V,
  a,a1,a2,b,b1,b2,c1,c2,n,k1,k2 for Real;

theorem
  Gen x,y & u,v,u1,v1 are_COrtm_wrt x,y & u,v,v1,w are_COrtm_wrt x,y
  implies u,v,u1,w are_COrtm_wrt x,y
proof
  assume
A1: Gen x,y;
  assume that
A2: u,v,u1,v1 are_COrtm_wrt x,y and
A3: u,v,v1,w are_COrtm_wrt x,y;
A4: Ortm(x,y,u),Ortm(x,y,v) // u1,v1 by A2;
A5: Ortm(x,y,u),Ortm(x,y,v) // v1,w by A3;
A6: u1,v1 // Ortm(x,y,u),Ortm(x,y,v) by A4,ANALOAF:12;
A7: now
    assume u<>v;
    then u1,v1 // v1,w by A1,A4,A5,Th6,ANALOAF:11;
    then
A8: u1,v1 // u1,w by ANALOAF:13;
     u1<>v1 implies thesis by A6,A8,ANALOAF:11;
    hence thesis by A3;
  end;
  u=v implies thesis by ANALOAF:9;
  hence thesis by A7;
end;
